Suppose that the distance of fly balls hit to the outfield (in baseball)is normally distributed with a mean of 250 feet and a standard deviation of50 feet.
a. If X = distance in feet for a fly ball, then X ~ _____(_____,_____)
b. If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 220 feet? Sketch the graph. Scale the horizontal axis X. Shade the region corresponding to the probability. Find the probability.
c. Find the 80th percentile of the distribution of fly balls. Sketch the graph, and write the probability statement.
answer
Find the 80th percentile of the distribution of fly balls
a. X ~ N(250, 50)
This means that the random variable X, which represents the distance in feet for a fly ball, follows a normal distribution with a mean of 250 feet and a standard deviation of 50 feet.
b. To find the probability that a randomly chosen fly ball traveled fewer than 220 feet, we need to calculate the area under the normal curve to the left of 220 feet.
To sketch the graph, draw a normal curve with the x-axis representing the distance in feet (X) and the y-axis representing the probability density. The mean (250) corresponds to the center of the curve.
To find the probability, we need to standardize the value of 220 feet using the z-score formula: z = (X - mean) / standard deviation.
z = (220 - 250) / 50
z = -2 / 50
z = -0.04
Next, we can use a standard normal distribution table or a calculator to find the probability associated with a z-value of -0.04. Alternatively, using a calculator such as Excel or a statistical software, we can directly calculate the probability using the formula for the cumulative distribution function of the normal distribution.
The shaded region represents the probability that the fly ball traveled fewer than 220 feet. Find the corresponding probability value using the standard normal distribution table or a calculator.
c. To find the 80th percentile of the distribution of fly balls, we need to find the distance (X) that corresponds to the 80th percentile. This is the point on the graph where 80% of the data falls to the left and 20% falls to the right.
To sketch the graph, draw a normal curve with the x-axis representing the distance in feet (X) and the y-axis representing the probability density. The mean (250) corresponds to the center of the curve.
To find the 80th percentile, we need to find the z-value that corresponds to the cumulative probability of 0.80. This can be done using a standard normal distribution table or a calculator.
Once we find the z-value, we can use the formula Z = (X - mean) / standard deviation to solve for X. Rearrange the formula to solve for X:
X = Z * standard deviation + mean
Plug in the appropriate values, and calculate X. This will give us the distance in feet corresponding to the 80th percentile of the distribution of fly balls.
a. Don't understand what you are seeking.
b. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Cannot graph on these posts.
c. Use the same table and look for Z score at .80. Use equation above. Again, cannot graph.